We present an efficient algorithm to find non-empty minimizers of a symmetric
submodular function over any family of sets closed under inclusion. This for
example includes families defined by a cardinality constraint, a knapsack
constraint, a matroid independence constraint, or any combination of such
constraints. Our algorithm make O(n3) oracle calls to the submodular
function where n is the cardinality of the ground set. In contrast, the
problem of minimizing a general submodular function under a cardinality
constraint is known to be inapproximable within o(n/logn) (Svitkina
and Fleischer [2008]).
The algorithm is similar to an algorithm of Nagamochi and Ibaraki [1998] to
find all nontrivial inclusionwise minimal minimizers of a symmetric submodular
function over a set of cardinality n using O(n3) oracle calls. Their
procedure in turn is based on Queyranne's algorithm [1998] to minimize a
symmetric submodularComment: 13 pages, Submitted to SODA 201